Difference between revisions of "2025 AMC 8 Problems/Problem 16"
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Five distinct integers from <math>1</math> to <math>10</math> are chosen, and five distinct integers from <math>11</math> to <math>20</math> are chosen. No two numbers differ by exactly <math>10</math>. What is the sum of the ten chosen numbers? | Five distinct integers from <math>1</math> to <math>10</math> are chosen, and five distinct integers from <math>11</math> to <math>20</math> are chosen. No two numbers differ by exactly <math>10</math>. What is the sum of the ten chosen numbers? | ||
| − | <math>\ | + | <math>\textbf{(A)}\ 95 \qquad \textbf{(B)}\ 100 \qquad \textbf{(C)}\ 105 \qquad \textbf{(D)}\ 110 \qquad \textbf{(E)}\ 115</math> |
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| + | ==Solution== | ||
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| + | Note that for no two numbers to differ by <math>10</math>, every number chosen must have a different units digit. To make computations easier, we can choose <math>(1, 2, 3, 4, 5)</math> from the first group and <math>(16, 17, 18, 19, 20)</math> from the second group. Then the sum evaluates to <math>1+2+3+4+5+16+17+18+19+20 = \boxed{\text{(C)\ 105}}</math>. | ||
Revision as of 20:48, 29 January 2025
Five distinct integers from 
to 
are chosen, and five distinct integers from 
to 
are chosen. No two numbers differ by exactly . What is the sum of the ten chosen numbers?
Solution
Note that for no two numbers to differ by , every number chosen must have a different units digit. To make computations easier, we can choose 
from the first group and 
from the second group. Then the sum evaluates to 
.
